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Logic
Negation - (~) the opposite of a statement; the negation would cancel out the original statement
Conjunction - (^) the combination or multiple statements; basically means “&”
Disjunction - (v) means “or”
Conditional - (p → q) a compound sentence that is formed by connecting two simple statements
using the words “IF....THEN...”
Converse - (q → p) interchanging the hypothesis and the conclusion of the original conditional
Inverse - (~p → ~q) denying both the hypothesis and the conclusion of the original conditional
Contrapositive - (~q → ~p) interchanging the hypothesis and the conclusion then denying both
of them from the original conditional
Bioconditional - (p ←> q) or [(p → q) ^ (q → p) IF AND ONLY IF
p
q
conj
p^q
disjunc
pvq
Condition
p→q
Converse
q→p
Inverse
~p → ~q
Contrapositive
~q → ~p
Biconditional
p ←> q
or
(p-->q)^(q-->p)
T
T
T
F
T
T
T
T
T
T
F
F
T
F
T
T
F
F
F
T
F
T
T
F
F
T
T
F
F
F
F
T
T
T
T
F
Law of Detachment (For conditionals!)
p→q
p
∴ q
Law of Contrapositive (ONLY SHOWS contrapositives!)
p→q
∴ ~q → ~p
Law of Modus Tollens (for contrapositives BUT IN THE FORM OF LAW OF DETACHMENT)
p→q
~q
∴ ~p
Law of Disjunctive Inference (If it’s not one, must be the other; works with disjunctives!)
pvq
~p
∴ q
DeMorgan’s Law (flip the sign, distribute the neg)
~ (p v q)
∴ ~p ^ ~ q
OR
~ (p ^ q)
∴ ~p v ~ q
Law of Syllogism (aka Chain Rule)
a→b
b→c
∴ a→c
Direct Proof - proving with the givens as it is, no assumptions
Indirect Proof - proving by assuming the opposite of what needs to be proven, then using the
givens until there is a contradiction in your conclusions and the given givens
EXAMPLE
Given: p → q
q→r
p
Prove: r
Direct
Indirect
Statements
Reasons
Statements
Reasons
1) p
Given
1) ~r
Assumption
2) p → q
Given
2) q → r
Given
3) q
Law of Detach (1, 2)
3) ~q
4) q → r
Law of Modus
Tollens (1, 2)
Given
4) p → q
Given
5) ∴ r
Law of Detach (3, 4)
5) ~p
Law of Contr (3, 4)
6) p
Given
7) r
Contradiction (5, 6)
**Note: Euler Diagrams son muy helpfulllllllllll**
Lines & Connerie porque yo no se que este unit was
Point - a dot with no shape or dimension that is used to indicate a specific location
Line - a connection of 2 points that go on forever; 1D ;)
Plane - a flat surface made up of 3 points that are noncollinear; 2D: Length & width
● in geo, parallelograms are used to represent planes
Collinear - when a point belongs on a line with another line
NonCollinear - points that do not belong on the same line
Ray - a line that only goes in ONE DIRECTION; terminates on one side but goes on forever on
the other
Coplanar - extends forever in all directions
Line Segment Addition Postulate (LSAP) - If A, B, C are collinear
& B is between A & C,
then AB + BC = AC
●
Contrapositive of LSAP - If AB + BC ≠ AC,
then A, B, C are not collinear
OR B is not between A & C (this is porque de DeMorgan’s)
Right Angle - an angle that is 90°
Linear Pair - two angles that are adjacent & supplementary together
Adjacent Angles - angles that are right next to each other; in contact with each other
Vertical Angles - angles that face each other and are created with 2 straight lines that intersect;
they’re congruent
Angle Addition Postulate - If a point lies on the interior of an angle, then that angle is the sum of
the two smaller angles with legs go through the given point
Addition Postulate - If equal quantities are added to equal quantities, the sums are equal.
If a=b & c=d, then a+c = b+d
Subtraction Postulate - If equal quantities are subtracted from equal quantities, the differences
are equal.
If a=b & c=d, then a-c = b-d
Multiplication Postulate - If equal quantities are multiplied by equal quantities, the products are
equal.
If a=b & c=d, then a*c = b*d
● Corollary: The Doubles Postulate - Doubles of equal quantities are equal
Division Postulate - If equal quantities are divided by nonzero equal quantities, the quotients are
equal
If a=b, c=d, c ≠ 0, d ≠ 0, then a/c = b/d
● Corollary: Halves Postulate - Halves of equal quantities are equal
Partitions Postulate - A whole is equal to the sum of it’s parts
Reflexive Postulate - A value is equal to itself?
Symmetric Postulate - If a=b, then b=a
Transitive Postulate - If ab = cd & cd = ef, then ab = ef
● Note: only used for values like lines & angles
Midpoint - Midpoint of a line segment is the point that is halfway between the endpoints of the
line segment.
●
If AB is a line segment and P is the midpoint, then AP = BP =
.
Angle Bisector - the ray (or the line) that divides an angle into two congruent angles. The
bisector will divide the angle equally in half.
Complementary Angle Theorem - angles that add up to be 90°
Supplementary Angles Theorem - angles that add up to 180°
Angle Theorems:
● If two angles are right angles, then they are congruent
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If two angles are complementary to the same angle, then they are congruent
If two angles are supplementary to the same angle, then they are congruent
If two angles are complementary to two congruent angles, then they are congruent
If two angles are supplementary to two congruent angles, then they are congruent
Angle Theorems can prove...
● Angle Congruence
● Supplementary Angles
● Complementary Angles
Parallel Lines
Parallel Lines - lines that are parallel; are the same distance apart from any point; never
intersect
Skew Lines - lines that don’t intersect each other but aren’t parallel because they’re
noncoplanar
Transversals - lines that intersect a system of lines
Corresponding Angles - http://www.mathnstuff.com/math/spoken/here/2class/260/trans5.gif
Alternate Interior Angles http://www.shodor.org/media/O/T/N/kZDViZjE5YzgyYzdhMjk2MDE5ZTVlOTdlN2IyZDc.gif
Alternate Exterior Angles http://www.mathwarehouse.com/geometry/angle/images/transversal/angles/alternate-exteriorangles.jpg
Perpendicular Lines - lines that intersect to create 90° angles
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Same-side interior angles of parallel lines are supplementary
A line perpendicular to one of two parallel lines is perpendicular to the other
Two lines perpendicular to the same line are parallel
Parallel Theorems:
● If two parallel lines are cut by a transversal, then the corresponding angles are equal
● If two parallel lines are cut by a transversal, then alternate interior angles are equal
● If two parallel lines are cut by a transversal, then alternate exterior angles are equal
● If two parallel lines are cut by a transversal, then consecutive interior angles are
supplementary
● If two parallel lines are cut by a transversal, then consecutive exterior angles are
supplementary
● If two parallel lines are cut by a transversal, then every pair of angles formed are either
equal or supplementary
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If the transversal is perpendicular to one of two parallel lines, then it is also
perpendicular to the other line
If two lines and a transversal form equal corresponding angles, then the lines are parallel
If two lines and a transversal for equal alternate interior angles, then the lines are
parallel
If two lines and a transversal form equal alternate exterior angles, then the lines are
parallel
If two lines and a transversal form consecutive interior angles that are supplementary,
then the lines are parallel
In a plane, if two lines are parallel to a third line, the two lines are parallel to each other
In a plane, if two lines are perpendicular to the same line, then the two lines are parallel
Parallel lines can prove...
● Angle congruence
● Supplementary Angles
Triangles (& polygons in general)
Angle Sum Theorem - The angle measures in any triangles add up to 180°
Corrolaries
● AA congruent to AA 3rd pair of angles congruent (IDK what this means...)
● Each angle of an equilateral triangle measures 60
● A triangle can have at most one right or one obtuse angle
● The acute angles of a right triangle are complementary
Polygons
Number of Sides
Name of Polygon
Number of Triangles
Sum of Interior
Angles
3
Triangle
1
180
4
Quadrilateral
2
180*2 = 360
5
Pentagon
3
180*3 = 540
6
Hexagon
4
180*4 = 720
7
Heptagon
5
180*5 = 900
8
Octagon
6
180*6 = 1080
9
Nonagon
7
180*7 = 1260
10
Decagon
8
180*8 = 1440
11
Hendecagon
9
180*9 = 1620
12
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Dodecagon
10
180*12 = 1800
If a polygon has n sides, sum of interior angles = 180(n - 2)
The sum of exterior angles for ALL polygons is 360°
Convex Polygon - a polygon such that no side extended cuts any other side or vertex; it can be
cut by a straight line in at most two points; diagonal cuts so that line only hits
areas IN the polygon
Concave Polygon - a polygon such that there is a straight line that cuts it in four or more points;
diagonal cuts so that line hits areas that are NOT in the polygon
Regular Polygon - polygons where the interior angles are equal to each other
● If a polygon has n sides, each interior angle is 180(n - 2)/n
Polygon
Measure of Each Int Angle of a Reg Polygon
Triangle
60
Quadrilateral
90
Pentagon
108
Hexagon
120
Heptagon
128.5714285714etc
Octagon
135
Nonagon
140
Decagon
144
Hendecagon
148.18(bar over “18”)
Dodecagon
150
Congruent Polygons - Polygons whose corresponding parts(sides+angles) are congruent
● Two polygons are congruent if their corresponding sides and angles are congruent
Congruent Triangle Postulates - are to be used to prove triangle congruence with congruent
corresponding parts
● SSS - If 3 sides of one triangle are congruent to 3 sides of another triangle, then the
triangles are congruent
● SAS - If 2 sides & the included angle of one triangle are congruent to 2 sides & the
included angle of another triangle, then the triangles are congruent
● ASA - If 2 angles and the included side of a triangle are congruent to 2 angles & the
included side of another triangle, then the triangles are congruent.
●
Corresponding Parts of Congruent Triangles are Congruent
You can use these four postulates/theorems to prove triangle congruence^^
Isosceles Triangle Theorem - If two sides of a triangle are congruent, then the angles opposite
those sides are congruent
● Converse: If two angles are congruent, then the sides opposite those angles are
congruent
● Corollary: The bisector of the vertex angle of an isosceles triangle is perpendicular to the
base at its midpoint
Equilateral Triangle Corollaries:
1 An equilateral triangle is also equiangular
2 An equilateral triangle has three 60° angles
3 An equiangular triangle is also equilateral
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AAS - If 2 angles & a side of a triangle are congruent of 2 angles & a side of another
triangle, then the two triangles are congruent
→ Note To Self: if u still dont get it, draw it out & talk out loud -- u’ll get it eventually
HL - If the hypotenuse & a leg of a right triangle are congruent to the hypotenuse & a leg
of another right triangle, then the two triangles are congruent
If two angles are supplementary to a pair of congruent angles, then the two angles are
congruent
Proofs with Overlapping Triangles & Double Triangles can be done with...
● SSS, ASA, SAS, AAS, HL → if they share a side, can be used for overlapping triangles
Also works for double triangles(but idk what doubles are...)
Exterior Angle Theorem - The measure of an exterior angle of a triangle is equal to the sum of
the measures of the two non-adjacent interior angles(remote interior
angles) of the triangle.
Angle Bisector Theorem - If a point is on the bisector of an angle, then it is equidistant from the
sides of the angle
● Converse: If a point is equidistant from the sides of an angle, then it is on the bisector of
the angle
Perpendicular Bisector Theorem - If a point lies on the perpendicular bisector of a segment,
then the point is equidistant from the endpoints of the segment
● Converse: If a point is equidistant from the endpoints of a segment, then the point lies on
the perpendicular bisector of the segment
Median - a line segment that comes out an interior angle of a triangle that bisects the side
opposite of it http://www.icoachmath.com/image_md/Median%20of%20a%20Triangle2.jpg
Altitude - the line segment drawn from any vertex of a triangle perpendicular to the line
containing the opposite side http://0.tqn.com/d/math/1/G/r/f/Altitude.gif
Angle Bisector - a line segment that cuts an angle into halves
http://mathworld.wolfram.com/images/eps-gif/AngleBisector_700.gif
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The median of an isosceles triangle is also the angle bisector of the angle it comes out
of
The altitude of an isosceles triangle is also the perpendicular bisector of the side it is in
contact with & is also the angle bisector of the vertex
The angle bisector of an isosceles triangle is also the perpendicular bisector of the side it
comes in contact with & is also the altitude of the triangle
In an isosceles triangle(& an equilateral triangle), the bisector of the vertex angle also
bisects the base making it the altitude & the median of the base
You can use the Isosceles Triangle and Equidistance Theorems(Def of Alt, Med, Angle Bis) to
prove....
● Angle congruence
● Side congruence
Inequality Properties:
1 If a>b, then a+c > b+c
2 If a>b, then a-c > b-c
3 If a>b & c>0, then ac > bc
4 If a>b & c<0, then ac < bc
5 If a>c & c>0, then a/c > b/c
6 If a>b & c<0, then a/c < b/c
7 If a>b & b>c, then a>c
8 If a<b & b<c, then a<c
9 If a>b & c≥0, then a+c > b+d
Exterior Angle Inequality Theorem - the exterior angle of a triangle is greater than any of the
remote interior angles
The larger angle is opposite to the larger side
● Converse: The larger the side is opposite to the larger angle
Triangle Inequality Theorem - In a triangle, the sum of the lengths of any two sides of a triangle
must be greater than the length of the third side
● Corollary - In a triangle, the length of any side of the triangle must be greater than the
difference between the other two sides
You can use the inequality properties/theorems to prove....
● Which angle/side is larger
Parallelograms/Quadrilaterals
Parallelogram - if both sides of opposite sides of a quadrilateral are parallel, then by definition
the quadrilateral is a parallogram
Properties of a Parallelogram:
1 Opposite sides are congruent
2 Opposite angles are congruent
3 Diagonals bisect each other
4 Opposite sides are parallel to each other(duh)
Parallelogram Theorems:
1 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a
parallelogram
2 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is
a parallelogram
3 If one pair of opposite sides are parallel & congruent in a quadrilateral, then the
quadrilateral is a parallelogram
4 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a
parallelogram
You can use the definition of a parallelogram & its properties to prove...
● Angle congruence
● Side/line congruence
● Sides are parallel
Methods for doing this:
● finding congruent triangles?
● use the givens....?
● Use alt int angles are congruent to prove parallel
Rectangle - a quadrilateral that is equiangular (all 90°)
Rectangle Theorem - A rectangle is equiangular
The diagonals of a rectangle are congruent
Properties that are EXCLUSIVE to the rectangle
● Diagonals are congruent
● All angles are right angles
● Equiangular
**All rectangles are parallel, converse doesn’t necessarily apply**
Ways to prove that a quad is a rectangle:
● Prove that it is a parallelogram with 1 right angle
● Prove it is an equiangular paralellogram
Area of a Rectangle:
b*h
Rhombus - a quadrilateral with 4 congruent sides
Ways to prove a quad is a rhombus:
● Prove it is a parallelogram with two consecutive congruent sides
● Prove it is a parallelogram with perpendicular diagonals (if the diagonals are congruent,
it’s a square)
● Prove it is a parallelogram with one diagonal bisecting the opposite angles through
which it is drawn
● Prove a quadrilateral is equilateral
Area of a Rhombus:
1) b*h
2) With the diagonals: ½(diagonal 1 * diagonal 2)
Square - a quadrilateral that is both equilateral & equiangular
Properties of a square:
1 A square is a rectangle in which all sides are congruent
2 A square’s angles are congruent/right
3 Diagonals of a square are congruent & perpendicular bisectors of each other
4 Diagonals of a square bisect the angles of a square
Theorems to prove that a quad is a square:
1 Prove that a quadrilateral is a rectangle in which two of the consecutive sides are
congruent
2 Prove it is a rhombus with one of whose angles is a right angle
Area of a Square:
1) side^2
2) With the diagonals: ½(diagonals)
If two lines are parallel, then all points on one line are equidistant from the other line
A line that contains the midpoint of one side of a triangle & is parallel to another side passes
through the midpoint of the third side.
If three parallel lines cut off congruent segments on one transversal, then they cut off congruent
segments on every transversal
(Parallel lines cut congruent segments on the transversal)
Trapezoid - a quadrilateral with one & ONLY ONE pair of parallel lines
● Diagonals of a trapezoid NEVER bisect each other
● If diagonals bisect, it’s not a trapezoid, it’s a rectangle
Types of Trapezoid:
1 Isosceles Trapezoids - base angles are congruent to its other base angle friend thing &
legs are congruent
http://image.tutorvista.com/content/feed/u826/isoculus%20trapezoid.JPG
2 Right Trapezoid - 2 right angles http://www.basicmathematics.com/images/Righttrapezoid.gif
The median (or mid-segment) of a trapezoid is parallel to each base and its length is one half
the sum of the lengths of the bases.
Area of a Trapezoid:
1) ½(b*h)
2) Height * median
Isosceles Trapezoids - a trapezoid where the legs are congruent which makes the base angles
congruent to their base angle friend
Properties of an Isosceles Trapezoid:
1 Legs are congruent
2 Base angles are congruent
3 Diagonals are congruent
4 Diagonals cut each other to be congruent(DO NOT BISECT EACH OTHER)
5 Opposite angles are supplementary
Isosceles Trapezoid Theorems
1 A trapezoid is isosceles if & only if the base angles are congruent
2 A trapezoid is isosceles if & only if the diagonals are congruent
3 If a trapezoid is isosceles, then the opposite angles are supplementary